Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.2.33

Chapter 4, Problem 4.2.33

Finding and Interpreting Mean, Variance, and Standard Deviation In Exercises 31–36, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.

Life on Other Planets Seventy-nine percent of U.S. adults believe that life on other planets is plausible. You randomly select eight U.S. adults and ask them whether they believe that life on other planets is plausible. The random variable represents the number who believe that life on other planets is plausible. (Source: Ipsos)

Verified step by step guidance

Step 1: Identify the parameters of the binomial distribution. The problem states that the probability of success (p) is 0.79 (79%), the probability of failure (q) is 1 - p = 0.21, and the number of trials (n) is 8.

Step 2: Calculate the mean (μ) of the binomial distribution using the formula: μ = n × p. Substitute the values of n and p into the formula.

Step 3: Calculate the variance (σ²) of the binomial distribution using the formula: σ² = n × p × q. Substitute the values of n, p, and q into the formula.

Step 4: Calculate the standard deviation (σ) by taking the square root of the variance: σ = √(σ²). Use the variance calculated in the previous step.

Step 5: Interpret the results. The mean represents the expected number of U.S. adults (out of 8) who believe that life on other planets is plausible. The standard deviation measures the typical deviation from the mean. To determine unusual values, use the range μ ± 2σ. Any value outside this range is considered unusual.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the random variable represents the number of U.S. adults who believe life on other planets is plausible, with a success probability of 0.79. Understanding this distribution is crucial for calculating the mean, variance, and standard deviation.

Mean, Variance, and Standard Deviation

The mean of a binomial distribution is calculated as the product of the number of trials and the probability of success, while the variance measures the spread of the distribution and is given by the product of the number of trials, the probability of success, and the probability of failure. The standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the mean. These statistics help summarize the distribution's characteristics.

Interpreting Results

Interpreting the results involves analyzing the calculated mean, variance, and standard deviation to understand the distribution's behavior. For instance, identifying unusual values can be done by examining how many standard deviations a value is from the mean. This interpretation helps in assessing the likelihood of certain outcomes and understanding the implications of the data in the context of the question.

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Textbook Question

Finding and Interpreting Mean, Variance, and Standard Deviation In Exercises 31–36, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.

Late for Work Thirty-one percent of U.S. employees who are late for work blame oversleeping. You randomly select 12 U.S. employees who are late for work and ask them whether they blame oversleeping. The random variable represents the number who are late for work and blame oversleeping. (Source: CareerBuilder)

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Textbook Question

Multinomial Experiments In Exercises 39 and 40, use the information below.

A multinomial experiment satisfies these conditions.

The experiment has a fixed number of trials n, where each trial is independent of the other trials.

Each trial has k possible mutually exclusive outcomes:

Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is

The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.

The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is

Genetics According to a theory in genetics, when tall and colorful plants are crossed with short and colorless plants, four types of plants will result: tall and colorful, tall and colorless, short and colorful, and short and colorless, with corresponding probabilities of , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless.

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Textbook Question

"Multinomial Experiments In Exercises 39 and 40, use the information below.

A multinomial experiment satisfies these conditions.

The experiment has a fixed number of trials n, where each trial is independent of the other trials.

Each trial has k possible mutually exclusive outcomes:

Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is

The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.

The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is

Genetics Another proposed theory in genetics gives the corresponding probabilities for the four types of plants described in Exercise 39 as , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless."

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Textbook Question

Discrete Variables and Continuous Variables In Exercises 13–18, determine whether the random variable x is discrete or continuous. Explain.

Let x represent the fitted hat sizes of members of a softball team.

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Textbook Question

Graphical Analysis In Exercises 9–12, determine whether the graph on the number line represents a discrete random variable or a continuous random variable. Explain your reasoning.

The distance a baseball travels after being hit

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Textbook Question

Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.

Civil Rights Fifty-nine percent of U.S. adults think that civil rights for Black Americans have improved during their lifetime. You randomly select seven U.S. adults. Find the probability that the number who think that civil rights for Black Americans have improved during their lifetime is (a) exactly one and (b) exactly five. (Source: Gallup)

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